Chaos Control Part II

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Chaos ControlPart II

• Amir massoud Farahmand
• SoloGen_at_SoloGen.net

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Review

• Why Chaos control?!
• THE BEGINNING WAS CHAOS!
• Chaos is Fascinating!
• Chaos is Everywhere!
• Chaos is Important!
• Chaos is a new paradigm shift in science!

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Review IIWhat is it?!

• Nonlinear dynamics
• Deterministic but looks stochastic
• Sensitive to initial conditions (positive Bol
  (Lyapunov) exponents)
• Strange attractors
• Dense set of unstable periodic orbits (UPO)
• Continuous spectrum

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Review IIIChaos Control Goals

• Stabilizing Fixed points
• Stabilizing Unstable Periodic Orbits
• Synchronizing of two chaotic dynamics
• Anti-control of chaos
• Bifurcation control

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Review IVChaos Control Methods

• Linearization of Poincare Map
• OGY (Ott-Grebogi-York)
• Time Delayed Feedback Control
• Impulsive Control
• OPF (Occasional Proportional Feedback)
• Open-loop Control
• Conventional control methods

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Chaos ControlConventional control

• Back-stepping
• A. Harb, A. Zaher, and M. Zohdy, Nonlinear
  recursive chaos control, ACC2002.
• Frequency domain methods
• Circle-like criterion to ensure L2 stability of a
  T-periodic solution subject to the family of
  T-periodic forcing inputs.
• M. Basso, R. Genesio, and L. Giovanardi, A. Tesi,
  Frequency domain methods for chaos control,
  2000.

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Chaos ControlConventional Chaotic

• Taking advantage of inherit properties of chaotic
  systems
• Periodic Chaotic systems are dense (according to
  Devaney definition)
• Waiting for the sufficient time, every point of
  the attractor will be visited.
• If we are sufficiently close to the goal, turn-on
  the conventional controller, else do nothing!
• T. Vincent, Utilizing chaos in control system
  design, 2000.

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Chaos ControlConventional Chaotic

• Henon map
• Stabilizing to the unstable fixed point
• Locally optimal LQR design
• Farahmand, Jabehdar, Stabilizing Chaotic Systems
  with Small Control Signal, unpublished

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Chaos ControlConventional Chaotic
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Chaos ControlConventional Chaotic
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Chaos Control

• Impulsive control of periodically forced chaotic
  system
• Z. Guan, G. Chen, T. Ueta, On impulsive control
  of periodically forced pendulum system, IEEE
  T-AC, 2000.

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Anti-Control of ChaosDefinitions and
Applications (I)

• Anti-control of chaos (Chaotification) is
• Making a non-chaotic system, chaotic.
• Enhancing chaotic properties of a chaotic system.

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Anti-Control of Chaos Definition and Applications
(II)

• Stability is the main focus of traditional
  control theory.
• There are some situations that chaotic behavior
  is desirable
• Brain and heart regulation
• Liquid mixing
• Secure communication
• Small control (Chaotification of non-chaotic
  system ? chaos control method (small control) ?
  conventional methods )

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Anti-Control of Chaos Discrete case (I)

• Suppose we have a LTI system. If we change its
  dynamic with a proper feedback such that it
• is bounded
• has positive Lyapunov exponent
• then we may have made it chaotic.
• We may use Marotto theorem to prove the existence
  of chaos in the sense of Li and Yorke.
• X. Wang and G. Chen, Chaotification via
  arbitrarily small feedback controls theory,
  methods, and applications, 2000.

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Anti-Control of Chaos Discrete case (II)
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Anti-Control of Chaos Discrete case (III)
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Anti-Control of ChaosContinuous case (I)

• Approximating a continuous system by its
  time-delayed version (Discrete map).
• Making a discrete dynamics chaotic is easy.
• It has not been proved yet!
• X. Wang, G. Chen, X. Yu, Anticontrol of chaos in
  continuous-time systems via time-delayed
  feedback, 2000.

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Anti-Control of ChaosContinuous case (II)
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Synchronization(I)

• Carrier Clock, Secure communication, Power
  systems and
• Formulation
• Synchronization
• Unidirectional (Model Reference Control)
• Mutual

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Synchronization(II)

• Linear coupling

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Synchronization(III)

• Drive-Response concept of Pecora-Carroll
• L.M. Pecora and T.L. Carol, Synchronization in
  chaotic systems, 1990.

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Synchronization of Semipassive systems (I)

• A. Pogromsky, Synchronization and adaptive
  synchronization in semipassive systems, 1997.
• Semipassive Systems

,

• Isidori normal form

• Control Signal

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Synchronization of Semipassive systems (II)

• Lemma Suppose that previous systems are
  semipassive with radially unbounded continuous
  storage function. Then all solutions of the
  coupled system with following control exist on
  infinite time interval and are bounded.

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Synchronization of Semipassive systems (III)

• Theorem I Assume that
• A1. The functions q, a, b are continuous and
  locally Lipschitz
• A2.The system is semipassive
• A3.There exist C2-smooth PD function V0 and
  that
• A4.The matrix b1b2 is PD
• A5.
• then there exist ? that goal of synchronization
  is achieved.

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Synchronization of Semipassive systems (IV)

• Lorenz system (Turbulent dynamics of the
  thermally induced fluid convection in the
  atmosphere)